Bayesian BWM - Probabilistic Group Best-Worst Method

Overview

Bayesian BWM generalises BWM to K decision makers and estimates the published hierarchical Dirichlet posterior by multi-chain MCMC. Outputs include posterior mean aggregate weights, 95% credible intervals, pairwise probabilities P(c_i>c_j), and the credibility-thresholded partial order; pairs below the threshold in both directions remain incomparable.

Output

Weight, higher is better

Data

Crisp, expert input required

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-12 criteria works best

Used for

Expert-driven decision making, MAGDM

How it works

1. 1

   Collect from each DM k: best B_k, worst W_k, BO vector A_B^k, OW vector A_W^k.

   $$\forall k\in \{1,\dots ,K\}:B_k,W_k\in \{1,\dots ,n\},A_B^k=(a_{B1}^k,\dots ,a_{Bn}^k),A_W^k=(a_{1W}^k,\dots ,a_{nW}^k)$$

   Mohammadi & Rezaei 2020, p.4 Sec.3.1

2. 2

   Hierarchical Dirichlet prior on aggregate weights w^agg and per-DM weights w^k.

   $$w^{agg}\sim \mathrm{Dir}(\mathbf{\alpha });w^k\mid w^{agg},\gamma \sim \mathrm{Dir}(\gamma w^{agg})\text{for }k=1,\dots ,K;\gamma \sim \mathrm{Gamma}(a,b)$$

   Mohammadi & Rezaei 2020, p.5 Eqs.(7)-(8)

3. 3

   Multinomial likelihood: OW counts proportional to w^k; reciprocals of BO entries proportional to w^k.

   $$A_W^k\mid w^k\sim \mathrm{Mult}\left({\sum }_j{a}_{jW}^k,w^k\right);(1/A_B^k)\mid w^k\sim \mathrm{Mult}\left({\sum }_{j}1/a_{Bj}^k,w^k\right)$$

   Mohammadi & Rezaei 2020, p.5 Eqs.(9)-(10)

4. 4

   Run MCMC (JAGS Gibbs sampler) for n_chains × n_iter iterations after n_burn burn-in. Collect posterior samples of (w^agg, w^1,...,w^K, γ).

   $$\{w^{agg,(s)},w^{k,(s)}{\}}_{s=1}^S\sim p\left(w^{agg},w^1,\dots ,w^K,\gamma |\{A_B^k,A_W^k{\}}_{k=1}^K\right)$$

   Mohammadi & Rezaei 2020, p.6 Sec.3.3

5. 5

   Posterior mean weights and 95% credible intervals; aggregate w^agg is the group weight vector.

   $${\hat{w}}_j^{agg}=\frac{1}{S}\sum _{s=1}^S{w}_j^{agg,(s)};C{I}_{95}(w_j^{agg})=[Q_{0.025},Q_{0.975}]$$

   Mohammadi & Rezaei 2020, p.7 Eqs.(13)-(14)

6. 6

   Credal ranking matrix: P(c_i > c_j) = (1/S) Σ_s 1[w_i^{agg,(s)} > w_j^{agg,(s)}]. Apply credibility threshold to produce partial order.

   $$P(c_i\succ c_j)=\frac{1}{S}\sum _{s=1}^{S}1\left[w_i^{agg,(s)}>w_j^{agg,(s)}\right];c_i{\succ }_{\tau }{c}_j⟺P(c_i\succ c_j)\ge \tau$$

   Mohammadi & Rezaei 2020, p.8 Eqs.(15)-(16)

Fits when / Look elsewhere when

Assumptions to verify

• Domain experts available
• Experts can express consistent comparisons

Edge cases and pitfalls

• Too few MCMC iterations → unstable posterior; check Gelman-Rubin R̂ < 1.1 per parameter.

• Highly inconsistent DM (large output ξ^k from non-Bayesian BWM) may dominate the group posterior unless reweighted; consider DM reliability weights.

Works with

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